Digital Signal Processing Reference
In-Depth Information
distributions that describe the random process as a collection of random variables are
all invariant to any time shift. As in the case of a random variable, the distribution for
a complex random process is defined as the joint distribution of real and imaginary
parts of the process.
For second-order stationarity, again we need to consider the complete characteriz-
ation using the pseudo-covariance function.
A complex random process
X
(
n
) is called
wide sense stationary
(WSS) if
EfX
(
n
)
g¼ m
x
,
is independent of
n
and if
E
{
X
(
n
)
X
(
m
)}
¼ r
(
nm
)
and it is called
second-order stationary
(SOS) if it is WSS and in addition, its pseudo-
covariance function satisfies and
E
{
X
(
n
)
X
(
m
)}
¼ p
(
nm
)
that is, it is a function of the time difference
n
2
m
.
Obviously, the two definitions are equivalent for real-valued signals and second-order
stationarity implies WSS but the reverse is not true. In [81], second-order stationarity
is identified as circular WSS and a WSS process is defined as an SOS process.
Let
X
(
n
) be a second-order zero mean stationary process. Using the widely-linear
transform for the scalar-valued random process
X
(
n
),
X
C
(
n
)
¼
[
X
(
n
)
X
(
n
)]
T
we
define the spectral matrix of
X
C
(
n
) as the Fourier transform of the covariance function
of
X
C
(
n
) [93], which is given by
C
(
f
)
P
(
f
)
P
(
f
)
C
(
f
)
C
C
(
f
)
W
F
{
E
{
X
C
(
n
)
X
C
(
n
)}}
¼
and where
C
(
f
) and
P
(
f
) denote the Fourier transforms of the covariance and pseudo-
covariance functions of
X
(
n
), that is, of
c
(
k
) and
p
(
k
) respectively.
The covariance function is nonnegative definite and the pseudo-covariance
function of a SOS process is symmetric. Hence its Fourier transform also satisfies
P
(
f
)
¼ P
(
2
f
). Since, by definition, the spectral matrix C
C
(
f
) has to be nonnegative
definite, we obtain the condition
2
jP
(
f
)
j
C
(
f
)
C
(
f
)
from the condition for nonnegative definiteness of C
C
(
f
). The inequality also states
the relationship between the power spectrum
C
(
f
) and the Fourier transform of a
pseudo-covariance function.
A random process is called second-order circular if its pseudo-covariance function
p
(
k
)
¼
0,
8k
a condition that requires the process to be SOS.
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